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determine the volume of the solid of revolution formed by rotating the region limited by the graphs y1= x+2; y2=x; x=0 and x=3 around the x axis

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Answer to a math question determine the volume of the solid of revolution formed by rotating the region limited by the graphs y1= x+2; y2=x; x=0 and x=3 around the x axis

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Hank
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Determine el volumen del sólido de revolución formado haciendo girar la región limitada por las gráficas y_1 = x + 2 ; y_2 = x ; x = 0 y x = 3 alrededor del eje x .

[SOLUTION]

1. Identificar las funciones dadas:
y_1 = x + 2
y_2 = x

2. Identificar los límites de integración:
x = 0
x = 3

3. Aplicar la fórmula del volumen mediante el método de los discos/lavadoras:
V = \pi \int_a^b [R(x)^2 - r(x)^2] \, dx

4. En este caso:
R(x) = x + 2
r(x) = x

5. Sustituir y resolver la integral:
V = \pi \int_0^3 [(x + 2)^2 - x^2] \, dx
V = \pi \int_0^3 [x^2 + 4x + 4 - x^2] \, dx
V = \pi \int_0^3 [4x + 4] \, dx
V = \pi \left[ 2x^2 + 4x \right]_0^3

6. Evaluar en los límites:
V = \pi \left[ 2(3)^2 + 4(3) - (2(0)^2 + 4(0)) \right]
V = \pi \left[ 18 + 12 \right]
V = 30\pi

Por lo tanto, el volumen del sólido de revolución es 30\pi .

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